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Full text of "Analytical Mechanics"

IMPULSE AND MOMENTUM                     259
205. Case II. Rough Contact. — When the plane is rough frictional forces come into play and change the tangential component of the momentum. Let F be the tangential force due to friction, N the normal force, and & the coefficient of friction; theu we have
X
= -mvn,
'= /    Ndt = -JT
Lt = /   F dt —  I   fjN dt = —
JQ              JQ
f*T'                   f*T'
L/= I    Fdt=  I    vN dt = — efj,mvn. JT            JT
But                          Lt+ L^ = mo I — mvt.
Therefore            m (vtf — vt) = — ra^ (1 + e) vn
and                      vtf =Vt—n(l+e) vn.
Substituting this value of vt' in the expression for tan which is obtained from Fig. 121, we get
vn                 evn
Eliminating vt between the last equation and the relation
tan a = ~ we obtain vn
e tan 0 = tan a - M (1 + e).                  (XV)
DISCUSSION. — When AC = 0, equation (XV) reduces to equation (XIV).
When M = oo , tan /3=— ooor/3 = — -; therefore the particle slides along
2i
the plane towards the left.   When e — 0 and tan a > p, tan /3 = oo and ft =  ~; therefore the particle slides along the plane towards the right.